From seven to eleven: Completely positive matrices with high cp-rank

Bomze, Immanuel M.; Schachinger, Werner; Ullrich, Reinhard

doi:10.1016/j.laa.2014.06.025

Cited by 29 publications

(52 citation statements)

References 11 publications

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“…Recently this conjecture has been disproven for n = 7, 8, 9, 10, 11 in [11] and for all n ≥ 12 in [12] (interestingly, it remains open for n = 6). Here we study our bounds on the examples of [11]. Although our bounds are not tight for the cp-rank, they are non-trivial and as such may be of interest for future comparisons.…”

Section: Examples Related To the Djl-conjecturementioning

confidence: 99%

Lower Bounds on Matrix Factorization Ranks via Noncommutative Polynomial Optimization

2019

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We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogues: the completely positive rank and the completely positive semidefinite rank. We study convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples.Keywords Matrix factorization ranks · Nonnegative rank · Positive semidefinite rank · Completely positive rank · Completely positive semidefinite rank · Noncommutative polynomial optimization Mathematics Subject Classification (2010) 15A48 · 15A23 · 90C22 1 Introduction Matrix factorization ranksA factorization of a matrix A ∈ R m×n over a sequence {K d } d∈N of cones that are each equipped with an inner product ·, · is a decomposition of the form A = ( X i ,Y j ) with X i ,Y j ∈ K d for all (i, j) ∈ [m] × [n], for some integer d ∈ N. Following [35], the

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Section: Examples Related To the Djl-conjecturementioning

confidence: 99%

Lower Bounds on Matrix Factorization Ranks via Noncommutative Polynomial Optimization

2019

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“…This conjecture has been proven for triangle free graphs in [9], for graphs which contain no odd cycle on 5 or more vertices in [8], for all graphs on 5 vertices which are not the complete graph in [19], for nonnegative matrices with a positive semidefinite comparison matrix (and any graph) in [4] and for all 5 × 5 completely positive matrices in [24]. However, Bomze et al [6] gave counterexamples to the Drew-JohnsonLoewy conjecture for real completely positive matrices of order seven through eleven.…”

Section: Cp-rank(g) = Max{cp-rank(a)|a Is Cp and G(a) = G}mentioning

confidence: 99%

Completely positive matrices over Boolean algebras and their CP-rank

Mohindru

2015

Special Matrices

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Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [︀ n 2 /4 ]︀ . In this paper, we prove this conjecture for n × n completely positive matrices over Boolean algebras (finite or infinite). In addition, we formulate various CP-rank inequalities of completely positive matrices over special semirings using semiring homomorphisms. Definition 1.4. [11] A semiring is a set S together with two operations ⊕ and ⊗ and two distinguished elements0, 1 in S with 0 ≠ 1, such that 1. (S, ⊕, 0) is a commutative monoid, 2. (S, ⊗, 1) is a monoid, 3. ⊗ is both left and right distributive over ⊕, 4. The additive identity 0 satisfies the property r ⊗ 0 = 0 ⊗ r = 0, for all r ∈ S.

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“…. , 11 were recently presented in [7]. For n ≥ 15 this conjecture is refuted by (8), and for n = 12, 13, 14 tighter lower bounds on p n also refute it [8].…”

Section: Perron-frobenius Perturbationsmentioning

confidence: 89%

The structure of completely positive matrices according to their CP-rank and CP-plus-rank

Bomze

Dickinson

Still

2015

Linear Algebra and its Applications

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We study the topological properties of the cp-rank operator cp(A) and the related cp-plus-rank operator cp + (A) (which is introduced in this paper) in the set S n of symmetric n × n-matrices. For the set of completely positive matrices, CP n , we show that for any fixed p the set of matrices A satisfying cp(A) = cp + (A) = p is open in S n \ bd (CP n ). We also prove that the set A n of matrices with cp(A) = cp + (A) is dense in S n . By applying the theory of semi-algebraic sets we are able to show that membership in A n is even a generic property. We furthermore answer several questions on the existence of matrices satisfying cp(A) = cp + (A) or cp(A) = cp + (A), and establish genericity of having infinitely many minimal cp-decompositions.

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From seven to eleven: Completely positive matrices with high cp-rank

Cited by 29 publications

References 11 publications

Lower Bounds on Matrix Factorization Ranks via Noncommutative Polynomial Optimization

Lower Bounds on Matrix Factorization Ranks via Noncommutative Polynomial Optimization

Completely positive matrices over Boolean algebras and their CP-rank

The structure of completely positive matrices according to their CP-rank and CP-plus-rank

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